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ON THE FREDHOLM PROPERTY OF A HYPERSINGULAR INTEGRAL OPERATOR IN A SCATTERING PROBLEM OF ELECTROMAGNETIC WAVES ON A SLAB COVERED WITH GRAPHENE
- Authors: Smirnov Y.G1, Tikhov S.V1
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Affiliations:
- Penza State University
- Issue: Vol 61, No 9 (2025)
- Pages: 1254–1271
- Section: INTEGRAL EQUATIONS
- URL: https://journals.rcsi.science/0374-0641/article/view/311971
- DOI: https://doi.org/10.7868/S3034503025090085
- ID: 311971
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Abstract
The paper focuses on a boundary-value problem for a system of two homogeneous Helmholtz equations with mixed boundary conditions arising in the theory of electromagnetic waves scattering on structures covered with two-dimensional materials. The problem is reduced to a boundary integral equation on the whole line involving a hypersingular integral operator. Introducing new variables and sought-for function, one passes to the integral equation on segment. To find an approximate solution of the obtained equation the collocation method using Fourier-Chebyshev series to present a solution is suggested; hypersingular integrals are calculated analytically. In the main part of the paper we prove the Fredholm property of the hypersingular operator involved in the integral equation under consideration.
About the authors
Yu. G Smirnov
Penza State University
Email: smirnovyug@mail.ru
Russia
S. V Tikhov
Penza State University
Email: tik.stanislav2015@yandex.ru
Russia
References
- Geim, A.K. The rise of graphene / A.K. Geim, K.S. Novoselov // Nat. Mater. — 2007. — № 6. — P. 183–191.
- Gorbach, A.V. Nonlinear graphene plasmonics: amplitude equation for surface plasmons / A.V. Gorbach // Phys. Rev. A. — 2013. — V. 87. — Art. 013830.
- Andreeva, V. Nonperturbative nonlinear effects in the dispersion relations for te and tm plasmons on two-dimensional materials / V. Andreeva, M. Luskin, D. Margetis // Phys. Rev. B. — 2018. — V. 98. — Art. 195407.
- Nonlinear optics of graphene and other 2D materials in layered structures / J.L. Cheng, J.E. Sipe, N. Vermeulen, C. Guo // J. Phys.: Photonics. — 2018. — V. 1, № 1. — Art. 015002.
- Song, J.H. Adaptive finite element simulations of waveguide configurations involving parallel 2D material sheets / J.H. Song, M. Maier, M. Luskin // Comput. Methods Appl. Mech. Eng. — 2019. — V. 351. — P. 20–34.
- Nonretarded edge plasmon-polaritons in anisotropic two-dimensional materials / D. Margetis, M. Maier, T. Stauber [et al.] // J. Phys. A: Math. Theor. — 2020. — V. 53, № 5. — Art. 055201.
- Smirnov, Yu. The nonlinear eigenvalue problem of electromagnetic wave propagation in a dielectric layer covered with graphene / Yu. Smirnov, S. Tikhov // Photonics. — 2023. — V. 10, № 5. — Art. 523.
- Integral equation analysis of plane wave scattering by coplanar graphene-strip gratings in the thz range / O.V. Shapoval, J.S. Gómez-Díaz, J. Perruisseau-Carrier [et al.] // IEEE Trans. Terahertz Sci. Technol. — 2013. — V. 3. — P. 666–674.
- Zinenko, T.L. Surface-plasmon, grating-mode, and slab-mode resonances in the h- and e-polarized thz wave scattering by a graphene strip grating embedded into a dielectric slab / T.L. Zinenko, A. Matsushima, A.I. Nosich // IEEE J. Sel. Top. Quant. — 2017. — V. 23. — P. 1–9.
- Смирнов, Ю.Г. Интегро-дифференциальные уравнения в задаче рассеяния электромагнитных волн на электрическом теле, покрытом графеном / Ю.Г. Смирнов, О.В. Кондырев // Дифференц. уравнения. — 2024. — Т. 60, № 9. — С. 1216–1224.
- Кауа, А.С. On the solution of integral equations with strongly singular kernels / А.С. Кауа, F. Erdogan // Q. Appl. Math. — 1987. — V. 45, № 1. — P. 105–122.
- Ervin, V. Collocation with chebyshev polynomials for a hypersingular integral equation on an interval / V. Ervin, E. Stephan // J. Comput. Appl. Math. — 1992. — V. 43, № 1. — P. 221–229.
- Lifanov, I.K. Hypersingular Integral Equations and Their Applications / I.K. Lifanov, L.N. Poltavskii, M.M. Vainikko. — 1st ed. — Boca Raton : CRC Press, 2003. — 416 p.
- Сетуха, A.B. Метод интегральных уравнений в математической физике: учеб. пособие / А.В. Ceryxa. — М. : Изд-во Моск. ун-та, 2023. — 316 с.
- Polyanin, A.D. Handbook of Linear Partial Differential Equations for Engineers and Scientists / A.D. Polyanin, V. E. Nazaikinskii. — 2nd ed. — New York : Chapman and Hall/CRC, 2016. — 1643 p.
- Тихов, С.В. Задача дифракции TE-волны на двумерном слое, покрытом графеном, с учётом нелинейности / С.В. Тихов // Модели, системы, сети в экономике, технике, природе и обществе. — 2024. — № 4. — С. 73–85.
- Shestopalov, Y.V. Logarithmic Integral Equations in Electromagnetics / Y.V. Shestopalov, Y.G. Smirnov, E.V. Chernokozhin. — Berlin; Boston : De Gruyter, 2000. — 128 p.
- Kober, H. Approximation by integral functions in the complex domain / H. Kober // Trans. Amer. Math. Soc. — 1944. — V. 56, № 1. — P. 7–31.
- Треногин, В.А. Функциональный анализ / В.А. Треногин. — М. : Наука, 1980. — 495 с.
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